CHAPTER XVIII — Critical Commentary & Notes

On Its Ninth Effect, Surpassing All the Others

Historical and Philological Commentary

  1. Intersecting Diagonals of the Regular Pentagon (Euclid XIII.8 and Euclid XIII.11): Proposition 8 of Book XIII proves that if two diagonals ac and be of an equilateral and equiangular pentagon abcde intersect at f, both diagonals cut one another in extreme and mean ratio ($ac : cf = cf : af$ and $be : ef = ef : bf$), and their greater segments equal the pentagon’s side ($cf = ef = ab$). Pacioli also cites Euclid XIII.11 (which proves that the side of the regular pentagon inscribed in a circle with rational diameter is the irrational line called a minor, $\dot{\varepsilon}\lambda\acute{\alpha}\sigma\sigma\omega\nu$, constructed from a fourth binomial and fourth apotome) because his Libellus (Tractatus I) and Summa extensively use Euclid XIII.8 and Euclid XIII.11 together to calculate all dimensions of the pentagon and dodecahedron.

Constantin Winterberg’s Commentary (1889 Edition)

Note 7 (Cap. XVIII): Ninth Effect — Diagonals of the Regular Pentagon (Euclid XIII.8) (Sachliche Anmerkungen #7, Cap. XVIII, p. 206, l. 11 from bottom)

The proof follows immediately from the similarity of the isosceles triangles $\triangle bca \sim \triangle baf$ formed by the two intersecting chords $ac$ and $be$ (meeting at $f$) of the regular pentagon:

\[\frac{ca}{ba} = \frac{ba}{af} \implies ba^2 = ca \cdot af,\]

and since $ba = bc = cf$, we have $cf^2 = ca \cdot af$, so the chord $ca$ is cut in extreme and mean ratio at $f$.

Anmerkung 7 (Cap. XVIII, S. 206): Neunter Effect — Diagonalen des regulären Fünfecks (Euklid XIII.8)

Der Beweis folgt aus der Aehnlichkeit der gleichschenkligen Dreiecke $\triangle bca$ und $\triangle baf$, wonach sich unmittelbar ergibt:

\[\frac{ca}{ba} = \frac{ba}{af}, \quad \text{d. h.} \quad ba^2 = ca \cdot af,\]

oder, da $ba = bc = cf$ ist: $cf^2 = ca \cdot af$.

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0028 (f. 6v–f. 6v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/018_cap_XVIII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/018_cap_XVIII.txt

  • Marginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.